In this section we discuss invariant cones
in the Lie algebra \(\mathcal {V}({\mathbb {S}}^1)\)  of smooth vector fields on the circle
and its (up to isomorphy unique) non-trivial central extension
\(\mathop {{\mathfrak {vir}}}\nolimits \) , the Virasoro algebra. For \(\mathcal {V}({\mathbb {S}}^1)\)  we show that, up to sign,
there is only one open invariant convex cone given by vector fields
of the form \(f \frac{\partial }{\partial \theta }\)  with \(f > 0\) .
As is well-known on the Lie algebra level, all unitary highest weight
representations of \(\mathcal {V}({\mathbb {S}}^1)\) , resp., the subalgebra of vector fields
for which \(f\)  is a finite Fourier polynomial, are trivial.
On the group level we show the closely related result that
all semibounded unitary representations of \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+\)  are trivial.
This is the main reason for the Virasoro algebra and the corresponding
simply connected group \(\mathop {{\rm Vir}}\nolimits \)  playing a more important role
in mathematical physics than \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+\)  itself
(cf. [1]}, [2]}, [3]}).
For \(\mathop {{\mathfrak {vir}}}\nolimits \)  we prove a convexity theorem for
adjoint and coadjoint orbits which provides complete
information on invariant cones and permits us to
determine the momentum sets of the unitary highest weight representations
of \(\mathop {{\rm Vir}}\nolimits \)  and to show that they are semibounded.
